The Italian Greyhound is an intelligent little dog, but they often use this to their own advantage to get their own way. They are known to be a little hard to housetrain, but with perseverance and a lot of understanding it is possible to teach an IG to do their business outside. Being sight hounds, it can be challenging to teach these dogs the "recall" command which is why their training has to start early and it has to be consistent bearing in mind that even the best trained IG might well ignore to respond to an owner if they have spotted something in the distance they find more interesting.
Italian Greyhounds are very sensitive dogs by nature and as such do not respond well to any sort of harsh correction or heavy handed training methods. They do answer well to positive reinforcement as long as their training is fair and consistent throughout their lives. However, it would be a mistake to let an IG off their leads anywhere near livestock or small animals no matter how well trained and responsive they are.
